COMPLEX RINGS AND QUATERNION RINGS

Abstract

In [4], complex rings C(R; −1), quaternion rings H(R;−1, −1) and octonion rings O(R; −1, −1) are studied for any ring R. For the real numbers R, C(R; −1) is the complex numbers, H(R; −1, −1) is the Hamilton’s quaternions and O(R; −1, −1) is the Cayley-Graves’s octonions. In view of progress of the quaternions, generalized quaternion algebras a,b F are introduced for commutative fields F and nonzero elements a, b ∈ F, and these quaternion algebras have been extensively studied as number theory. In this paper, we use H(F; a, b) instead of a,b F . For a division ring D and nonzero elements a, b in the center of D, we introduce generalized complex rings C(D; a) and generalized quaternion rings H(D; a, b), and study the structure of these rings. We show that, if 2 = 0, that is, the characteristic of D is not 2, then H(D; a, b) is a simple ring and C(D; a) is a simple ring or a direct sum of two simple rings. Main purpose of this paper is to study structures of these simple rings. We also study the case of 2 = 0.